Generator of local symmetries

Question

Let us only consider classical field theories in this discussion.

Noether's theorem states that for every global symmetry, there exists a conserved current and a conserved charge. The charge is the generator of the symmetry transformation. Concretely, if $\phi \to \phi + \varepsilon^a\delta_a \phi$ is a symmetry of the action, then there exists a set of conserved charges $Q_a$, such that $$ \{Q_a, \phi\} = \delta_a \phi $$

If the symmetry is a gauge symmetry, then no such conserved current or charge exists (although certain constraints can be obtained via Noether's second theorem). However, I have the following question

Is there a quantity that, like $Q_a$ generates the gauge transformation?

This quantity need not be conserved. To be precise, if an action is invariant under local transformations $\phi \to \phi + [\varepsilon^a(x) \delta_a] \phi$. Here, heuristically, the gauge transformation might look like $[\varepsilon^a \delta_a] \phi \sim d \varepsilon + \varepsilon \phi + \cdots$ where $d \varepsilon$ is heuristically some derivative on $\varepsilon$. Does there exist a quantity such that $$ \{ Q_{\varepsilon}, \phi \} = [ \varepsilon^a \delta_a] \phi $$ I know there exist such charges in some theories, and I have explicitly computed them as well. I was wondering if there is a general formalism to construct such quantities?

PS - I have some ideas about this, but nothing quite concrete.

Answer 1

Let us here assume that the classical theory is given by a Hamiltonian (as opposed to a Lagrangian) formulation, so that we have a Poisson bracket $\{\cdot,\cdot\}_{PB}$ (and so that we can discuss whether or not the generators form a Poisson algebra or not).

A generic theory will have constraints (and corresponding Lagrange multipliers). The constraints are not a result of any Noether procedure, but are typically found systematically via the Dirac-Bergmann recipe when performing a (singular) Legendre transformation from the Lagrangian to the Hamiltonian formulation.

The constraints are divided into first and second-class constraints, say $G_a$, and $\chi_{\alpha}$, respectively. The Dirac conjecture states that the first-class constraints $G_a$ generate gauge symmetries

$$\delta_{\varepsilon}=\varepsilon^a\{G_a,\cdot\}_{PB},$$

cf. e.g. Ref. 1, which also lists a counterexample. Ref. 1 defines infinitesimal gauge transformations$^1$ $\delta_{\varepsilon}$ as quasisymmetry transformations that depend on infinitesimal gauge parameters $\varepsilon^a(x)$, which in turn are allowed to depend arbitrarily on the spacetime point $x$. Such spacetime-dependent (or so-called local) quasisymmetry transformations are the starting point of Noether's second theorem. See e.g. this Phys.SE post, where the corresponding Noether charges are discussed.

The presence of second-class constraints $\chi_{\alpha}$ imply that one must ultimately replace the Poisson bracket $\{\cdot,\cdot\}_{PB}$ with the pertinent Dirac bracket $\{\cdot,\cdot\}_{DB}$.

References:

1. M. Henneaux and C. Teitelboim, Quantization of Gauge Systems, 1994; Section 1.2.

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$^1$ Concerning the notion of gauge symmetry, see also e.g. this Phys.SE post.